Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the stanar response questions. Write your answers clearly! Inclue enough steps for the graer to be able to follow your work. Don t skip limits or equal signs, etc. Inclue wors to clarify your reasoning. Do first all of the problems you know how to o immeiately. Do not spen too much time on any particular problem. Return to ifficult problems later. If you have any questions please raise your han an a proctor will come to you. You will be given exactly 90 minutes for this exam. This is a practice exam. The actual exam may iffer significantly from this practice exam because there are many varieties of problems that can test each concept. ACADEMIC HONESTY Do not open the exam booklet until you are instructe to o so. Do not seek or obtain any kin of help from anyone to answer questions on this exam. If you have questions, consult only the proctor(s). Books, notes, calculators, phones, or any other electronic evices are not allowe on the exam. Stuents shoul store them in their backpacks. No scratch paper is permitte. If you nee more room use the back of a page. Anyone who violates these instructions will have committe an act of acaemic ishonesty. Penalties for acaemic ishonesty can be very severe. All cases of acaemic ishonesty will be reporte immeiately to the Dean of Unergrauate Stuies an ae to the stuent s acaemic recor. I have rea an unerstan the above instructions an statements regaring acaemic honesty:. SIGNATURE Page of
Integrals FORMULA SHEET Derivatives Volume: Suppose A(x) is the cross-sectional area of the soli S perpenicular to the x-axis, then the volume of S is given by (sinh x) = cosh x x Inverse Trigonometric Functions: (cosh x) = sinh x x V = b a A(x) x x (sin x) = x x (csc x) = x x Work: Suppose f(x) is a force function. The work in moving an object from a to b is given by: W = b x = ln x + C x tan x x = ln sec x + C a f(x) x sec x x = ln sec x + tan x + C a x x = ax ln a + C for a Integration by Parts: u v = uv v u x (cos x) = x x (tan x) = x (sec x) = + x x (cot x) = x x + x If f is a one-to-one ifferentiable function with inverse function f an f (f (a)) 0, then the inverse function is ifferentiable at a an (f ) (a) = f (f (a)) Hyperbolic an Trig Ientities Hyperbolic Functions sinh(x) = ex e x cosh(x) = ex + e x csch(x) = sinh x sech(x) = cosh x tanh(x) = sinh x cosh x coth(x) = cosh x sinh x Page of cosh x sinh x = cos x + sin x = sin x = ( cos x) cos x = ( + cos x) sin(x) = sin x cos x sin A cos B = [sin(a B) + sin(a + B)] sin A sin B = [cos(a B) cos(a + B)] cos A cos B = [cos(a B) + cos(a + B)]
Stanar Response Questions. Show all work to receive creit. Please BOX your final answer.. Consier the region in the xy-plane above the curve y = x 3, below the line y = 8, an to the right of the y-axis. (a) (9 points) Sketch this region, an compute the volume of the soli obtaine by rotating this region aroun the y-axis (b) (9 points) Assume the soli in part (a) is a storage tank (with all lengths in feet), full of a liqui weighing 4 pouns per cubic foot. Compute the work (in ft-lbs) neee to pump all the liqui to the feet above of the top of the tank. Page 3 of
. Evaluate the following limits. x (a) (6 points) lim x 0 ln(sec(3x)) ln x (b) (6 points) lim x log 5 (x) (c) (6 points) lim x (x) 3/x Page 4 of
3. Fin the erivative for each of the following functions. (a) (4 points) y = sin(6xe 3x ) (b) (4 points) g(t) = e t + t (c) (5 points) g(x) = 7 ln(x 3x+) () (5 points) y(x) = x 7 sin(x) Page 5 of
4. Evaluate the following integrals. (a) (6 points) x with x <. x (b) (6 points) x x with x. (c) (6 points) x + 4 x 3 + x x Page 6 of
5. Solve the following initial value problems. (a) (9 points) y t = tet, y(0) = (b) (9 points) y (x) = e x +, y(0) = 4, y (0) = 5 Page 7 of
Multiple Choice. Circle the best answer. No work neee. No partial creit available. 6. (7 points) Mark the correct answer for the integral given by A. 0 B. C. a D. a a a sin(sin(x 3 ))x. 7. (7 points) Compute the improper integral an mark the correct answer A. 0 B. C. 3 D. 3 6 3 x 3 x 8. (7 points) Compute the ln(sinh(7x)) an mark the correct answer x A. 7 cosh x sinh x sinh x B. cosh x cos x C. sinh x D. 0 Page 8 of
9. (7 points) Mark the correct simplification for the expression A. B. e 3 ln(x ) 4 ln e (x ) e 3 ln(x ) 4(x ) e3 ln(x ) ln e (x )4 C. e 3 (x ) D. (x ) 0. (7 points) If f(3) =, f (3) = 7, then (f ) () is A. 3 B. 7 C. 7 D. 7. (7 points) The partial fraction ecomposition of can be written in the form A. A =, B =. B. A = 3, B = 3. C. A = 3, B = 3. A x + D. A = 3(x ), B = (x + ). B x + where 3 (x )(x + ) Page 9 of
. (7 points) It took 600 J of work to stretch a spring from its natural length of 3 m to a length of 8 m. The spring constant k is A. 30 B. 640 C. 00 D. 8 3. (7 points) Let P (t) grams be the population of bacteria in a tank at t hours. Suppose the population oubles every 3 hours, an P () =. Fin P (t). A. P (t) = t B. P (t) = (t+)/3 C. P (t) = t/3 D. P (t) = e t 4. (7 points) When evaluating the integral x 3 x + 36 x, which of the following woul be the best substitution for x? A. x = 36 sin t B. x = 6 sin t C. x = 6 tan t D. x = 6 sec t Page 0 of
Congratulations you are now one with the exam! Go back an check your solutions for accuracy an clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be hane in. Please have your MSU stuent ID reay so that is can be checke. DO NOT WRITE BELOW THIS LINE. Page Points Score 3 8 4 8 5 8 6 8 7 8 8 9 0 Total: 53 No more than 50 points may be earne on the exam. Page of